Astrometric and timing effects of gravitational waves from localized sources

Abstract

The extremely high precision of current radio interferometric observations demands a better theoretical treatment of secondary effects in the propagation of electromagnetic signals in variable gravitational fields. Such fields include those of oscillating and precessing stars, stationary or coalescing binary systems, and colliding galaxies. Especially important is the problem of propagation of light rays in the field of gravitational waves emitted by a localized source of gravitational radiation. A consistent approach for a complete and exhaustive solution of this problem is developed in the present paper in the first post-Minkowskian and quadrupole approximation of general relativity. This approximation is linear with respect to the universal gravitational constant G and accounts for the static monopole, spin, and time-dependent quadrupole moments of an isolated system. We demonstrate for the first time that the equations of light propagation in the retarded gravitational field of an arbitrary localized source emitting quadrupolar gravitational waves can be integrated exactly in closed form. The influence of the gravitational field under consideration on the light propagation is examined not only in the wave zone but also in cases when light passes through the intermediate and near zones of the source. We reproduce the known results of integration of equations of light rays, both in a stationary gravitational field and in the field of plane gravitational waves, establishing the relationship between our new formalism and the simplified approaches of other authors. Explicit analytic expressions for light deflection and integrated time delay (Shapiro effect) are obtained accounting for all possible retardation effects and arbitrary relative locations of the source of gravitational waves, the source of light rays, and the observer. Coordinate dependent terms in the expressions for observable quantities are singled out and used for physically meaningful interpretation of observable quantities. It is shown that the ADM and harmonic gauge conditions can both be satisfied simultaneously outside the source of gravitational waves. Such ADM-harmonic coordinates are extensively used in the present paper. Their use drastically simplifies the integration of light propagation equations and the equations for the motion of light source and observer in the gravitational field of the source of gravitational waves, leading to the unique interpretation of observable effects. The two limiting cases of small and large values of impact parameter d are elaborated in more detail. It is proved that leading order terms for the effect of light deflection in the case of small impact parameter depend neither on the radiative part $(\ensuremath{\sim}1/d)$ of the gravitational field nor on the intermediate $(\ensuremath{\sim}{1/d}^{2})$ zone terms, confirming a previous result in the literature. The main effect rather comes from the near zone $(\ensuremath{\sim}{1/d}^{3})$ terms. This property of strong suppression of the influence of gravitational waves on the propagation of light rays makes much more difficult any direct detection of gravitational waves by VLBI or pulsar timing techniques, in contrast with previous claims by other authors. We also present a thorough-going analytical treatment of time delay and bending of light in the case of large impact parameter. This exploration essentially extends previous results regarding propagation of light rays in the field of a plane monochromatic gravitational wave. Explicit expressions for Shapiro effect and deflection angle are obtained in terms of the transverse-traceless (TT) part of the space-space components of the metric tensor. We also discuss the relevance of the developed formalism for interpretation of radio interferometric and timing observations, as well as for data processing algorithms for future gravitational wave detectors.